You have five pirates, ranked from
5 to 1 in descending order. The top pirate has the right to propose how
100 gold coins should be divided among them. But the others get to vote
on his plan, and if fewer than half agree with him, he gets killed. How
should he allocate the gold in order to maximize his share but live to
enjoy it? (Hint: One pirate ends up with 98 percent of the gold.)

Here is my solution. First, it must be clarified that if the top
pirate is killed, then the next highest ranking pirate gets to propose
his plan. So, if pirate #5 proposes a plan, and then the remaining
pirates think it suck, then pirate #5 dies, and pirate #4 is then forced to make a proposal as to a new plan (leaving out pirate #5 for obvious reasons).

So, it turns out that its easiest to work backwards! If only pirate
#2 and pirate #1 remain, pirate #1 should never vote for the plan if
pirate #2 gives himself anything. In fact, if pirate #2 proposes a
0/100 division, then pirate #1 is actually being merciful if he allows
pirate #2 to live, even with no gold.

So if we add pirate #3 to the list (in the 3-pirate case), pirate #2
will never vote against a plan, because it guarantees that he lives,
and if there were only 2 pirates left, he would have no chance of
keeping any gold anyway. So, pirate #3 would propose a 100/0/0 split,
with pirate #2 voting for him, and pirate #1 voting against him, which
would pass the proposed gold division. (Hence, we know that it can
never drop to below a 3 pirate scenario.)

Let’s add pirate #4 to the mix. Pirate #4 can determine that pirate
#3 will never vote for him, because if pirate #4 died, pirate #3 can
get 100 pieces of gold. Can pirate #4 get pirate #1 and pirate #2 on
his side? Well, pirate #4 can propose that he keeps 98 pieces of gold,
and pirate #1 and pirate #2 keep 1 piece of gold each. Pirates #1 and
pirates #2 like this, as they will end up with no gold if pirate #3
gets to make his proposal.

It turns out that pirate #5 can make the same deal as pirate #4.
Pirate #5 gives himself 98 pieces of gold, and pirates #1 and 2 1 piece
of gold each. Pirates 1 and 2 are getting the same deal as with pirate
4, so its up to them to let pirate 5 get the 98 pieces of gold or
pirate 5 die and pirate 4 get the 98 pieces of gold. (Either way is
equivalent to pirates 1 and 2 in terms of gold distribution, but since
the dynamics of whether pirate 1 or 2 has anything against pirate 5 to
the point where they’d like to kill him and instead give the 98 gold to
pirate 4 is undefined here, we’ll assume “intelligence” pirates who
want the game to end sooner, so they’ll just agree to pirate #5’s plan.)